On Aug 28, 5:02 pm, John H Palmieri <jhpalmier...@gmail.com> wrote: > > - Are there (useful) cases where the direct product of a subcategory > > of Sets does not coincide with the cartesian product on the > > underlying sets? > > Not that I can think of right now.
The category of schemes. Not really a subcategory of sets, but even forgetting the algebraic part, the point set of a product scheme is not the cartesian product of the corresponding point sets. > > - There could be some ambiguity for tensor product which is often > > used as an alias for cartesian product for graphs, crystals, ... > > But those are not subcategories of VectorSpaces/Modules. So it > > would not be an issue to have a similar alias in Sage. > > That sounds okay to me. The ambiguity here comes because the cartesian product for graphs, etc gives these categories a monoidal structure, as does the tensor product in the category of vector spaces. This "categorical tensor product" does not need to be either a categorical product nor a coproduct (but sometimes it is). > I think that "cartesian_product" is a good, unambiguous (I think) name > for the set construction. "direct_sum" and "direct_product" are maybe > okay, while definitely "coproduct" and possibly "product" should be > reserved for the category-level operation. +1 I would say that each category has its own operations defined with the usual name, eg cartesian_product for sets or vector spaces, direct_product for groups, monoids or algebras, and so on, the "category level operations" should be consistently named (product, coproduct, limit, colimit, ...) and mapped to whatever the corresponding construction is in each particular category. Back to the original problem, "direct_sum" should be defined for vector spaces, but not for algebras, or if defined for algebras through coercion, then return a vector space, and not an algebra. Cheers Javier PS: Yes, yes, I'll do the reviews. Busy right now finishing a paper with deadline (sunday). Will focus on this as soon as I am done. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
